Experiment 1
We assessed the CE of multiple predator species and the extent to which NCE contribute toward the production of an emergent MPE by conducting an experiment in plastic tubs (58.4 × 42.5 × 15.2 cm). Tubs were filled with 13 L of filtered water and stocked with 150 g of washed sweet gum (Liquidambar styraciflua) leaves collected from a natural pond. Each tub received 30 newly hatched Bufo tadpoles (Gosner stages 23–28). We independently manipulated the abundance (zero vs. one) of two freeswimming predator species (newts and Anax) as is typical of studies examining CEs of predators and emergent MPEs. The average mass (±SE) of adult newts in the experiment was 1.36 g ± 0.06 and the average mass of the lateinstar Anax larvae in the experiment was 1.14 g + 0.07. Densities of Bufo tadpoles (125/m^{2}), larval Anax (4/m^{2}), and adult newts (4/m^{2}) are reflective of densities found in natural ponds (Appendix S2).
Our study included two additional treatments that are not typically included in studies examining MPE; one where an Anax could consume Bufo in the presence of a caged newt and one where a newt could consume Bufo in the presence of a caged Anax. These treatments allowed us to measure the extent to which one predator species alters the foraging rate of a second predator species without physically interacting with the other predator or their prey. Cages for predators were 0.27 L opaque plastic cups with pin holes punched in the side which provided an opportunity for predators and prey to sense the presence of another predator species via chemical cues while preventing physical interactions or visual detection. All tubs received a cage so as to not confound the occurrence of a caged predator with the occurrence of a cage.
Treatments were randomly assigned to tubs and predator individuals were randomly assigned to tubs of the appropriate treatment. Prey were added to tubs 2 h before predators. We emptied tubs and counted the number of surviving Bufo 24 h after the addition of predators. Each of the six treatments was replicated once within each of seven blocks (see Appendix S2 for description of blocking structure).
We estimated instantaneous mortality rates (# tadpoles that die/individual/24 h) of Bufo in each tub as the absolute value of the ln proportion of Bufo that survived to the end of the experiment. We used the absolute value so that mortality risk is represented by a positive value. This approach assumes that neither reproduction nor migration occurs and that the number of Bufo present declines in an exponential manner (Gotelli 2008); all of which are reasonable assumptions in our study. We performed analyses on instantaneous mortality rates because the multiplicative risk model predicts the combined CE of multiple predators by summing the independent effects of predators on instantaneous mortality rates (i.e., summing the independent effects of predators on logtransformed estimates of percent survival) when these assumptions are true (Wilbur and Fauth 1990; Billick and Case 1994; Sih et al. 1998). In other words, the multiplicative risk model predicts the combined mortality risk imposed by multiple predators (μ_{na}) on their prey when the predators do not alter the mortality risk imposed by each other as:
 (Model(1))
where μ_{n} is the instantaneous mortality risk (i.e., CE) imposed by newts, μ_{a} is the instantaneous mortality risk (i.e., CE) imposed by Anax, and μ_{b} is the background instantaneous mortality risk of prey in the absence of predators. Model 1 can be modified to predict the combined mortality risk imposed by multiple predators (μ_{na}) on their prey when predators modify the mortality risk of prey to other predators:
 (Model(2))
where j refers to the NCE that nonphysical interactions with Anax has on the instantaneous mortality risk imposed by newts, k refers to the NCE that physical interactions with Anax (e.g., interference competition or intraguild predation) has on the instantaneous mortality risk imposed by newts, p refers to the NCE that nonphysical interactions with newts has on the instantaneous mortality risk imposed by Anax, and q refers to the NCE that physical interactions with newts has on the instantaneous mortality risk imposed by Anax. Our mechanistic model 2 is similar to a mechanistic model described by Rudolf (2008), however, we employ an additive approach to modifying mortality risk rather than a multiplicative approach (i.e., μ_{n} + j + k rather than μ_{n}jk).
We present model 2 with an additive approach because we present a statistical approach below that allows us to efficiently estimate all the parameter estimates (and their standard errors) in model 2, evaluate whether each parameter is significantly different than zero, and compare the observed combined mortality risk to those expected from several null models that make different assumptions about j and k while only conducting a single analysis of variance (ANOVA) with planned contrasts. Furthermore, because j, k, p, and q refer to the extent to which one predator alters the CE of another predator via a NCE, the influence of CE and NCE is expressed in similar units using the additive approach which allows us to directly evaluate how much each contributes to the combined effect of multiple predators on prey mortality.
In our study, we assume that k = q = 0 because (1) intraguild predation is unlikely between these two predator species and (2) it is not possible to experimentally separate the influences of j and k from the influences of p and q (although we can estimate j and p, we cannot estimate k and q) when the removal of any potential behavioral influence by one predator likely also removes the influence of physical interference (although k and q could be estimated for intraguild predation if appropriate treatments that manipulate the densities of each predator to reflect the influence of intraguild predation are conducted). Nonetheless, our design can suggest the importance of k and q if our models, which assume they are zero, fail to adequately predict the combined effect of multiple predators.
We tested eight hypotheses regarding the CE and NCE of multiple predators on a common prey. The first four hypotheses (contrasts 1–4) evaluated whether the CE of each predator species (i.e., μ_{n} and μ_{a}) and the NCE of each predator species on the CE of the other predator species (i.e., j and p) are different from zero. The last four hypotheses (contrasts 5–8) evaluated whether the observed prey mortality risk in the presence of multiple predators is different from that expected by (i) model 2 when j = p = 0, (ii) model 2 when j = 0 and p reflects the observed effect on μ_{a}, (iii) model 2 when j reflects the observed effect on μ_{n} and p = 0, and (iv) model 2 when j reflects the observed effect on μ_{n} and p reflects the observed effect on μ_{a}. We compared the observed prey mortality risk in the presence of multiple predator species to each of these four models to identify the model with the parameter estimates that best predicts the combined effect of multiple predators. To test these hypotheses we performed eight planned contrasts in conjunction with a oneway ANOVA comparing instantaneous mortality rates among all six treatments using PROC MIXED in SAS (Cary, NC) (Table 1). A block effect was excluded from the ANOVA model because it accounted for little variation in the data.
Table 1. Planned contrasts to test eight hypotheses pertaining to the impact of predators on the instantaneous mortality rates of Bufo terrestrisContrast  CNLA  LA  LALN  LN  CALN  None 

(μ_{a}, p, μ_{b})  (μ_{a}, μ_{b})  (μ_{n}, j, μ_{a}, p, μ_{b})  (μ_{n}, μ_{b})  (μ_{n}, j, μ_{b})  (μ_{b}) 


1. Does Anax affect the mortality rate of B. terrestris relative to background mortality rates?H_{o} = LA − None = (μ_{a} + μ_{b}) − (μ_{b}) = μ_{a} = 0  0  +1  0  0  0  −1 
2. Do newts affect the mortality rate of B. terrestris relative to background mortality rates?H_{o} = LN − None = (μ_{n} + μ_{b}) − (μ_{b}) = μ_{n} = 0  0  0  0  +1  0  −1 
3. Does the nonconsumptive effect of Anax alter the consumptive effect of newts on B. terrestris?H_{o} = CALN − LN = (μ_{n} + j + μ_{b}) − (μ_{n} + μ_{b}) = j = 0  0  0  0  −1  +1  0 
4. Does the nonconsumptive effect of newts alter the consumptive effect of Anax on B. terrestris?H_{o} = CNLA−LA = (μ_{a} + p + μ_{b}) − (μ_{a} + μ_{b}) = p = 0  +1  −1  0  0  0  0 
5. Does model 2 adequately predict the combined effect of multiple predators when we assume that the NCE of each predator on the CE of the other predator is unimportant (i.e., j = p = 0)? This is thetraditional test of model 1.H_{o} = (LALN + None) − (LA + LN) = ((μ_{n} + 0 + μ_{a} + 0 + μ_{b}) + μ_{b}) − ((μ_{a} + μ_{b)} + (μ_{n} + μ_{b})) = 0  0  −1  +1  −1  0  +1 
6. Does model 2 adequately predict the combined effect of multiple predators when we assume that NCE produced by nonphysical interactions with Anax is important (i.e., j ≠ 0) while the NCE produced by nonphysical interactions with newts is unimportant (i.e., p = 0)?H_{o} = (LALN + None) − (LA + CALN) = ((μ_{n} + j + μ_{a} + 0 + μ_{b}) +μ_{b}) − ((μ_{a} + μ_{b}) + (μ_{n} + j + μ_{b})) = 0  0  −1  +1  0  −1  +1 
7. Does model 2 adequately predict the combined effect of multiple predators when we assume that NCE produced by nonphysical interactions with newts is important (i.e., p ≠ 0) while the NCE produced by nonphysical interactions with Anax is unimportant (i.e., j = 0)?H_{o} = (LALN + None) − (CNLA + LN) = ((μ_{n} + 0 + μ_{a} + p + μ_{b}) + μ_{b}) − ((μ_{a} + p + μ_{b}) + (μ_{n} + μ_{b})) = 0  −1  0  +1  −1  0  +1 
8. Does model 2 adequately predict the combined effect of multiple predators when we assume that the NCEs produced by nonphysical interactions with both predators are important (i.e., p ≠ 0 and j ≠ 0)?H_{o} = (LALN + None) − (CNLA + CALN) = ((μ_{n} + j + μ_{a} + p + μ_{b}) + μ_{b}) − ((μ_{a} + p + μ_{b}) + (μ_{n} + j + μ_{b})) = 0  −1  0  +1  0  −1  +1 
Contrasts 5–8 evaluated whether there was a statistical interaction between the effect of newt presence and the effect of Anax presence; the presence of a statistical interaction between these two effects on estimates of instantaneous mortality rates means that the observed combined mortality rate is different from that expected by the null model (Sih et al. 1998; VanceChalcraft et al. 2004). Contrasts 5–8 differed from each other by changing the treatments used in the contrast to represent the effects of newt presence and the effect of Anax presence, but all four contrasts included the treatments with no predators to incorporate the influence of background mortality rates (where mortality risk is represented by μ_{b}) and the treatment with lethal Anax and newts to provide the observed estimate for the combined effect of multiple predators (where mortality risk is represented by model 2). For example, contrast 5 includes the treatment with lethal newts alone (where mortality risk is represented by μ_{n} + j + μ_{b} and j = 0 because no Anax are present) and the treatment with lethal Anax alone (where mortality risk is represented by μ_{a} + p+ μ_{b} and k = 0 because no newts are present) in order to evaluate the null model outlined in hypothesis 5 – this is the traditional analysis used to evaluate model 1 or model 2 where it is assumed that j = k = p = q = 0. Contrast 6, on the other hand, includes the treatment with lethal newts and caged Anax (where mortality risk is represented by μ_{n} + j + μ_{b} and j represents the actual NCE of caged Anax on the CE of newts) and the treatment with lethal Anax alone (where mortality risk is represented by μ_{a} + p + μ_{b} and p = 0 because no newts are present) in order to evaluate the null model outlined in hypothesis 6.
Rejecting hypothesis 5 means that at least some of the NCE of predators on the CE of other predators (i.e., j, k, p, and/or q) are important parameters to include in the model, whereas failing to reject it suggests that they are unimportant. Failing to reject one of the hypotheses 6–8 indicates that we have identified a model that includes an important NCE produced by nonphysical interactions (either j and/or p) that is sufficient to predict the combined effect of multiple predators. Rejecting hypotheses 5–8 indicates that NCE produced by physical interactions (either k and/or q) are important parameters to include in the model even though it may not be possible to estimate them. We used four LSMESTIMATE statements employing the same treatment weights as the first four contrasts in PROC MIXED to obtain the least square estimates (and standard error) of μ_{n}, μ_{a}, j, and p. An additional LSMESTIMATE statement estimated μ_{b} by assigning a weight of zero to all treatments except for the predatorfree control which received a weight of one. Predicted effects for each of the models were derived by summing parameter values assumed (although many are calculated) for each model.
There are several advantages to the statistical approach we employ in comparison to other studies which employ a mechanistic approach to evaluate how other predators alter the mortality risk imposed by other predators (e.g., Crumrine and Crowley 2003; Rudolf 2008; Crumrine 2010). First, our approach utilizes all the data together in a single analysis (rather than multiple analyses with different subsets of the data) which enhances statistical power for all hypotheses because the greater number of independent replicates in the analysis reduces the estimate of the error MS used to test hypotheses. Second, by conducting a single analysis with the data there is no variation in the error structure for different hypothesis tests that would otherwise occur if one performed multiple analyses that utilized different, but partially overlapping data sets. Third, our approach reduces the potential for committing type I errors given that fewer hypothesis tests are actually being performed. Fourth, all the parameter estimates in the mechanistic model are directly estimated from a single statistical mode rather than performing lots of analyses on different subsets of the data. Indeed, a great advantage of our approach is that it directly relates least square estimates from our statistical analysis to the parameter estimates in our mechanistic model and so there is no disconnect between the mechanistic model and the statistical model. The connection between the statistical and mechanistic model is important because our statistical model can lead to more accurate parameter estimates by allowing us to account for other sources of variation that may influence mortality risk. For example, we could use body size estimates for individual predators in each experimental unit as a covariate in the statistical analysis so that the least square parameter estimates are adjusted to account for differences in mortality risk due to differences in predator size. Fifth, the approach we outline does not require one to match individual replicates from different treatments together in order to estimate parameters or predicted mortality risks which has been a concern in other studies. Sixth, our approach allows us to evaluate whether observed mortality risk in the presence of multiple predators is statistically different from multiple null models without biasing the error structure of the analysis. Others have treated different null models as different treatments in an analysis and considered the predicted values for each treatment as independent which biases the estimate of the error MS used to test the hypotheses because the different predictions in each treatment are not really independent. Although our contrasts are not necessarily independent of each other (and we can correct for this with the false discovery rate), the data in the analysis are completely independent.
We report probability values for each contrast that were adjusted to control the false discovery rate (P_{fdr}; Verhoeven et al. 2005) (See Appendix S3 for original ANOVA results) as the contrasts are not orthogonal. We present unadjusted Pvalues in Appendix S3. All statistical analyses were conducted in SAS Enterprise Guide (SAS 2010).